Trigonometry — Finding the area of a polygon inscribed in a circle

Question

A regular nonagon is inscribed in a circle with a radius spanning 5 cm. Find the perimeter of P and area A of the nonagon.

Answer 1

The area of a regular Nonagon can be calculated using the basic formula A = 9/4 * a^2 * cot(20°), where a represents the length of one side. You can obtain 2 * a/2 directly from the two right triangles formed by taking the radii at two edges on the Nonagon and splitting that triangle in half, using basic Trigonometry.

Answer 2

For regular n-gons, we can consider the area as the sum of n identical isosceles triangles whose paired lengths will equal the radius and whose other angle is 360°/n. The equal angles will be (180° - 360°/n)/2 =

90° - 180°/n. Then, bisect the central angle part of the triangle. It's angle will then be 180°/n; thus, we have two right triangle with angles 180°/n, 90° - 180°/n, and 90°, and the hypotenuse has length r. Then, the lengths of the other 2 sides are r sin 180°/n (this is half of the side length of the n-gon) and r cos 180°/n.

Then, the area of each triangle is 1/2 r sin 180°/n r cos 180°/n, so the area of the original triangle made up of two of these triangles is r sin 180°/n r cos 180°/n. Recall that sin 2θ = 2 sinθ cosθ. Then, the area of the triangle is 1/2 r² sin 360°/n. Thus, to get the perimeter and area of the original n-gon, multiply the side length and area of these triangles by n, and we get

perimeter = 2nr sin 180°/n

area = n/2 r² sin 360°/n

If you substitute π for 180°, noting that sin x/x has limit 1 as x goes to 0, we see that the limiting perimeter will be 2n r π/n = 2π r, and the area of the n-gon is n/2 r² * 2π/n = πr² .

Now, for the nonagon, which has 9 sides, the perimeter is 2nr sin 180°/n = 2 * 9 * 5 sin 180°/9 = 90 sin 20°, and the area is n/2 r² sin 360°/n = 9/2 * 5² * sin 40° = 225/2 sin 40°

Calculating these, we get the perimeter 90 sin 20°is equal to 30.7818128993102

The circumference 225/2 sin 40° = 72.3136060897357

Note that the circumference of the circle, 2π r, is 2 * π * 5 = 10π =31.4159265358979

The area of the circle, πr² , is π5² = 25π = 78.5398163397448